Nuclear Processes

Nuclear reactions are completely independent of chemical reactions, in the sense that nuclear energies are several orders of magnitude larger. Thus an atom which has undergone a nuclear reaction reacts the same chemically as an identical atom which has not.

There are four major types of nuclear reaction:

  1. fission, the splitting of a nucleus into two "daughter" nuclei with the help of a neutron. The most common fission reaction is
    n + 235U -> 141Ba + 92Kr + 3n,
    where the superscripts indicate the nucleon numbers (denoted by A). Equal to the sum of the numbers of protons and neutrons in the nucleus, A is also approximately equal to the atomic mass, when mass is measured in atomic mass units (1.66055 * 10-27 kg). When sufficient amounts of Uranium are present, the extra neutrons may initiate additional fission reactions, leading to a chain reaction.

    This reaction produces approximately 188 MeV of excess kinetic energy, accounting for its explosive potential. It is a prime example of the the scaling relationship mentioned in the last section: smaller sources are associated with larger energies.

  2. fusion of two "parent" nuclei into one daughter nucleus. The fusion reaction at the heart of most stars begins with the reaction
    p + p -> 2H + e+ + ν + .42 MeV.
    Here ν denotes a neutrino: a neutral and nearly massless counterpart to the electron. Neutrinos interact via the weak force: trillions pass through you each second without any reaction at all with the atoms in your body! The weak force is qualitatively different from
    electromagnetism, gravity and the strong force. It is called the weak force because it is weaker by a factor of 10000 than the strong force, although it is not the weakest: gravity is 1036 times weaker. Gravity only appears to be so strong because of the sizes of the sources of the gravitational field: planets, stars and galaxies.

    The e+ denotes a positron: a particle of antimatter. An antiparticle is identical to its ordinary "partner" (in this case an electron) in every way except its charge and helicity (whether its spin is parallel or antiparallel to its momentum). The positron is created in order to conserve charge, and the neutrino is necessary to "balance" the creation of the positron.

    Electrons, positrons and neutrinos are collectively called leptons. There were no leptons on the left hand side of the fusion reaction, so there must be none on the right: the positron, with lepton number -1, plus the neutrino, with lepton number 1, add up to 0 leptons.

    Leptons cannot interact via the strong force, although those which have electric charge can interact electromagnetically. And in perhaps one of the most bizarre mysteries in physics, only left-handed leptons (those whose spin is parallel to their momentum) can interact via the weak force. Right-handed leptons, with negative helicity, are weakly sterile, as are right-handed quarks. And to date, no one has ever detected a right-handed neutrino, although there is evidence that the neutrino has a very small mass, and so right-handed neutrinos should exist.

    Something else interesting happened in this reaction: a proton turned into a neutron. What actually happened was that one up quark changed into a down quark. The Standard Model of Particle Physics models the interactions between particles as an exchange of gauge bosons. For instance, the electromagnetic force between two electrons is caused by the exchange of a photon between them; the photon is the gauge boson for electromagnetism. Similarly, the gluon is the gauge boson for the strong force, and the W+, W- and Z0 particles are the gauge bosons for the weak force. So in this reaction, one up quark emitted a W+, which changed it from an up quark with electric charge +2/3 into a down quark with electric charge -1/3, and then the W+ decayed into a positron and a neutrino. And we know it was a left-handed up quark: because it cannot interact weakly, a right-handed up quark will always be an up quark.
    Quarks and leptons are fermions, which obey the Pauli Exclusion Principle: no two can have the same quantum numbers and be in the same place at the same time. Bosons do not suffer this same restriction. In addition, fermions have spin 1/2 while bosons have spin 1.
    The extra energy that is released in this reaction comes from the fact that the combined mass of the products is lower than the combined mass of the reactants. This mass difference is equivalent to energy through Einstein's famous equation
    E = m c2
    and comes about because the binding energy of the resulting Deuterium nucleus (2H) is negative. Differences in binding energies also accounts for the excess energy released in fission.

  3. neutron capture (used to create radioactive isotopes): in this reaction the nuclear charge Z is unchanged, A increases by one and the number of neutrons (denoted N) increases by one (note that N always is equal to A - Z).
  4. nuclear decay, which occurs whenever a nucleus is in an energy state which is not the lowest possible for its nucleon number.

Note that fission and neutron capture start out the same; the difference in fission is that the excited nucleus (with the extra neutron) is too unstable to stay together. In neutron capture, the new nucleus is simply radioactive, that is, it decays in one of the following modes:

  1. alpha decay: emission of an alpha particle (equivalent to a Helium nucleus, which consists of two protons and two neutrons). This is most common for elements with Z > 82, since it provides the greatest energy loss per nucleon.

  2. beta decay, which can be
    the transformation of a neutron into a proton (a down quark into an up quark), with the emission of an electron for charge conservation, and an antineutrino for lepton conservation; or

    the transformation of a proton into a neutron plus a positron and a neutrino.

    Beta decay occurs in those situations in which alpha decay would leave the nucleus less stable than it was before.
    Neutrinos and antineutrinos are very difficult to detect: experiments can typically detect only one in 1015! Their existence was first predicted to account for missing energy and momentum in beta decays.
  3. gamma decay: high energy photon emission in the nuclear version of x-rays. This occurs when an unstable nucleus cannot alpha or beta decay because of nuclear conservation rules which are outside the scope of this text. Note that all nuclear processes must conserve energy, charge, momentum, angular momentum and lepton number.

These decays are called by the common name of radioactivity. Isotopes, which are atoms with the same Z but different A and N, are designated by AelementZ (ie., 235U92). Typically, one unstable nucleus will decay into another unstable nucleus, over and over in a decay series until an ultra-stable nucleus, usually 206, 207 or 208Pb82, is reached as the end product.

Three such series occur naturally, one beginning with 232Th90, and the others beginning with 238U92 and 235U92. The Thorium Series is

232Th90 ->
α + 228Ra88 ->

β + 228Ac89 ->

β + 228Th90 ->

α + 224Ra88 ->

α + 220Rn86 ->

α + 216Po84 ->

α + 212Pb82 (99.987% of the time) ->
β + 212Bi83

β + 216At85 (.013% of the time) ->

α + 212Bi83,
with the 212Bi83 ->
α + 208Tl81 (33.7% of the time) ->
β + 208Pb82

β + 212Po84 (66.3% of the time) ->

α + 208Pb82.
The branches in the decay series occur when two decay modes are possible with (usually) differing probabilities. We can never tell for certain when any given nucleus will decay, and if a branch is possible, which branch it will take when it does decay. For this reason, radioactive decay rates are based on large samples (numbers on the order of Avogadro's number).

We suggest the construction of the 238U 92 series as an exercise for the reader. It includes the following intermediate isotopes: 210Bi, 214Bi, 234Pa, 210Pb, 214Pb, 210Po, 214Po, 218Po, 226Ra, 222Rn, 230Th, 234Th, 210Tl and 234U, and ends with 206Pb. In all of the beta decays Z -> Z + 1. Note that there is a branch in this series.

The following applet knows all of the alpha and electron-beta decay modes for the currently known elements. It will give you an isotope and the decay mode, and you must compute Z, A and N for the decay product. All answers must be exact! You may find it useful to refer to the Periodic Table of the Elements.

You need a Java-capable browser to be able to use the applets. If they do not work with your Windows system, download the Java VM (Virtual Machine) for your version of Windows at the download section at

In a large population of unstable nuclei, the decays occur apparently at random, but since the decay rate is proportional to the number of nuclei present, the overall decay rate is described by an exponential function

D = D0 2 - t / τ.
Here D0 is the initial amount of the parent nucleus, D is the amount left at time t, and since the base for the exponential is 2, τ is a half-life (the time it takes for D to be 1/2 of D0). The half life is proportional to the stability of the excited nucleus: more stable nuclei have longer half lives, etc. It is analogous to the time constant in an RC circuit: it is a characteristic value for a given isotope. D may be measured in units of mass, but is often measured in Curies (Ci, corresponding to about 1g of Ra, or about 3.7 * 1010 decays/s). Half lives from the Thorium Series are
232Th901.405 * 1010 years
228Ra885.75 years
228Ac896.15 hours
228Th901.9131 years
224Ra883.66 days
220Rn8655.6 seconds
216Po840.145 seconds
212Pb8210.64 hours
212Bi8325 / 60.55 minutes
216At850.33 milliseconds
208Tl813.053 minutes
212Po8445.1 seconds
Biological organisms are affected by exposure to radioactivity. Low velocity alpha particles are more destructive since they tend to interact more often than high velocity beta particles. A mitigating effect occurs when radioactive atoms are metabolized within the body. The radioactive decay mode provides a "physical" half-life, and the metabolic process provides a "biological" half-life. The two act as parallel paths for elimination of the radioactivity, and in analogy with parallel resistors, the effective half life is then
1 / τeffective = 1 / τphysical + 1 / τbiological.
This is the last section of the text. We hope you have enjoyed learning about the applications of physics to some of the topics in biology and chemistry which interest you most, and that you will find both the information you have learned and the thought processes you've developed useful in your future academic and professional career. We wish you the best of luck!

©2013, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.

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